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TRANSCRIPT
Math 104 -‐ Yu
Balancing Masses
• Along the line, mass m1 at point x1, mass m2 at x2. • Archimedes’ Law of Lever: rod will be balanced if
m1d1 = m2d2
d1 d2
m1(x� x1) = m2(x2 � x)
m1x�m1x1 = m2x2 �m2x
(m1 +m2)x = m1x1 +m2x2x =
m1x1 +m2x2
m1 +m2
Center of Mass:
Moment of the system about the origin
Total Mass
Math 104 -‐ Yu
Center of Mass Along a Line • A system of n masses: m1, m2, ..., mn
• Center of mass:
• Moment of the system about the origin: measure the tendency of the system to rotate about the origin.
x =m1x1 + · · ·+mnxn
m1 + · · ·+mn
M0 = m1x1 + · · ·+mnxn
Math 104 -‐ Yu
Balancing on the Plane • For a planar region, it can rotate about either x or y-‐axis.
So there are two moments:
• Total Mass:
• The center of mass is:
• Moment about x-axis: M
x
= m1y1 + · · ·mn
y
n
.
• Moment about y-axis: M
y
= m1x1 + · · ·mn
x
n
.
M = m1 + · · ·+mn
x =M
y
M
, y =M
x
M
Math 104 -‐ Yu
Center of Mass of a 2D Region • To compute moments and total mass of a region with a
given density (mass to area raPo), we parPPon it into strips and do a Riemann sum approximaPon.
Math 104 -‐ Yu
Center of Mass of a 2D Region • If the density only depends on the x coordinate, total
mass is given by
• The moments are:
M =
Z b
a�(x)[f(x)� g(x)]dx
M
y
=
Zb
a
x�(x)[f(x)� g(x)]dx
M
x
=
Zb
a
1
2[f(x) + g(x)]�(x)[f(x)� g(x)]dx
=1
2
Zb
a
�(x)[f(x)2 � g(x)2]dx
x = x
y =f(x) + g(x)
2˜dm = �dA
= �(x)[f(x)� g(x)]dx
Math 104 -‐ Yu
Centroid • If the density is constant the formula simplify:
• In this case the value of the density is irrelevant. We also call the center of mass the centroid of the region.
x =M
y
M
=
Rb
a
x[f(x)� g(x)]dxRb
a
[f(x)� g(x)]dx
y =M
x
M
=
Rb
a
[f(x)2 � g(x)2]dx
2Rb
a
[f(x)� g(x)]dx
Total area of the region A
Math 104 -‐ Yu
Examples 1. Find the center of mass of a thin plate between the x-axis and y =
2/x
2, 1 x 2, if the density is �(x) = x
2.
2. Find the centroid of an isosceles triangle whose base is on the x-axis,
�1 x 1 and whose height is 3.
Math 104 -‐ Yu
Examples 3. Find the center of mass of a plate of constant density given by the region
between y = x� x
2and y = �x.